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Existence results for singular three point boundary value problems on time scales. (English) Zbl 1069.34012

Summary: We prove the existence of a positive solution for the three point boundary value problem on time scale 𝕋 given by

y ΔΔ +f(x,y)=0,x(0,1]𝕋,y(0)=0,y(p)=yσ 2 (1),

where p(0,1)𝕋 is fixed and f(x,y) is singular at y=0 and possibly at x=0, y=. We do so by applying a fixed point theorem due to J. A. Gatica, V. Oliker and P. Waltman [J. Differ. Equations 79, 62–78 (1989; Zbl 0685.34017)] for mappings that are decreasing with respect to a cone. We also prove the analogous existence results for the related dynamic equations y ΔΔ +f(x,y)=0, y Δ +f(x,y)=0, and y Δ +f(x,y)=0 satisfying similar three point boundary conditions.

34B10Nonlocal and multipoint boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
39A12Discrete version of topics in analysis